Abstract

In a diamond, the mechanical vibration-induced strain can lead to interaction between the mechanical mode and the nitrogen-vacancy (NV) centers. In this work, we propose to utilize the strain-induced coupling for the quantum non-demolition (QND) single phonon measurement and memory in a diamond. The single phonon in a diamond mechanical resonator can be perfectly absorbed and emitted by the NV centers ensemble (NVE) with adiabatically tuning the microwave driving. An optical laser drives the NVE to the excited states, which have much larger coupling strength to the mechanical mode. By adiabatically eliminating the excited states under large detuning limit, the effective coupling between the mechanical mode and the NVE can be used for QND measurement of the single phonon state. Under realistic experimental conditions, we numerically simulate the scheme. It is found that the fidelity of the absorbing and emitting process can reach a much high value. The overlap between the input and the output phonon shapes can reach 98.57%.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2017 (3)

Y. Yanay and A. A. Clerk, “Shelving-style phonon-number detection in quantum optomechanics,” New J. Phys. 19, 033014 (2017).
[Crossref]

E. R. MacQuarrie, M. Otten, S. K. Gray, and G. D. Fuchs, “Cooling a mechanical resonator with nitrogen-vacancy centres using a room temperature excited state spin-strain interaction,” Nat. Commun. 8, 14358 (2017).
[Crossref] [PubMed]

K. Cai, R. X. Wang, Z. Q. Yin, and G. L. Long, “Second-order magnetic field gradient-induced strong coupling between nitrogen-vacancy centers and a mechanical oscillator,” Sci. China Phys. Mech. 60, 070311 (2017).
[Crossref]

2016 (9)

D. A. Golter, T. Oo, M. Amezcua, I. Lekavicius, K. A. Stewart, and H. Wang, “Coupling a Surface Acoustic Wave to an Electron Spin in diamond via a Dark State,” Phys. Rev. X 6, 041060 (2016).

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016).
[Crossref] [PubMed]

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016).
[Crossref] [PubMed]

Y. Ma, Z. Q. Yin, P. Huang, W. L. Yang, and J. F. Du, “Cooling a Mechanical Resonator to Quantum Regime by heating it,” Phys. Rev. A 94, 053836 (2016).
[Crossref]

Y. Liu, F. Kong, F. Shi, and J. Du, “Detection of radio-frequency field with a single spin in diamond,” Sci. Bull. 61, 1132 (2016).
[Crossref]

P. Arrangoiz-Arriola and A. H. Safavi-Naeini, “Engineering interactions between superconducting qubits and phononic nanostructures,” Phys. Rev. A 94, 063864 (2016).
[Crossref]

Q. Hou, W. Yang, C. Chen, and Z. Yin, “Electromagnetically induced acoustic wave transparency in a diamond mechanical resonator,” J. Opt. Soc. Am. B 33, 2242–2250 (2016).
[Crossref]

X. Wang, A. Miranowicz, H. R. Li, and F. Nori, “Method for observing robust and tunable phonon blockade in a nanomechanical resonator coupled to a charge qubit,” Phys. Rev. A 93, 063861 (2016).
[Crossref]

T. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Sci. Bull. 61, 163–171 (2016).
[Crossref]

2015 (4)

M. J. A. Schuetz, E. M. Kessler, G. Giedke, L. M. K. Vandersypen, M. D. Lukin, and J. I. Cirac, “Universal Quantum Transducers Based on Surface Acoustic Waves,” Phys. Rev. X 5, 031031 (2015).

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041002 (2015).

Z. Yin, N. Zhao, and T. Li, “Hybrid opto-mechanical systems with nitrogen-vacancy centers,” Sci. China Phys. Mech. 58, 050303 (2015).
[Crossref]

Z. Yin, W. L. Yang, L. Sun, and L. M. Duan, “Quantum network of superconducting qubits through opto-mechanical interface,” Phys. Rev. A 91, 012333 (2015).
[Crossref]

2014 (6)

Y. Tao, J. M. Boss, B. A. Moores, and C. L. Degen, “Single-crystal diamond nanomechanical resonators with quality factors exceeding one million,” Nat. Commun. 5, 3638 (2014).
[Crossref] [PubMed]

C. Galland, N. Sangouard, N. Piro, N. Gisin, and T. J. Kippenberg, “Heralded single-phonon preparation, storage, and readout in cavity optomechanics,” Phys. Rev. Lett. 112, 143602 (2014).
[Crossref] [PubMed]

M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekström, G. Johansson, and P. Delsing, “Propagating phonons coupled to an artificial atom,” Science 346, 207–211 (2014).
[Crossref] [PubMed]

V. M. Stojanovic, M. Vanevic, E. Demler, and L. Tian, “Transmon-based simulator of nonlocal electron-phonon coupling: a platform for observing sharp small-polaron transitions,” Phys. Rev. B 89, 144508 (2014).
[Crossref]

P. Ovartchaiyapong, K. W. Lee, B. A. Myers, and A. C. B. Jayich, “Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator,” Nat. Commun. 5, 4429 (2014).
[Crossref] [PubMed]

O. P. deSaNeto, M. C. deOliveira, F. Nicacio, and G. J. Milburn, “Capacitive Coupling of Two Transmission Line Resonators Mediated by the Phonon Number of a Nanoelectromechanical Oscillator,” Phys. Rev. A 90, 023843 (2014).
[Crossref]

2013 (4)

Z. Q. Yin, T. C. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88, 033614 (2013).
[Crossref]

N. Bar-Gill, L.M. Pham, A. Jarmola, D. Budker, and R.L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
[Crossref] [PubMed]

S. Saito, X. Zhu, R. Amsüss, Y. Matsuzaki, K. Kakuyanagi, T. Shimo-Oka, N. Mizuochi, K. Nemoto, W. J. Munro, and K. Semba, “Towards realizing a quantum memory for a superconducting qubit: storage and retrieval of quantum states,” Phys. Rev. Lett. 111, 107008 (2013).
[Crossref]

S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, “Phonon-induced spin-spin interactions in diamond nanostructures: application to spin squeezing,” Phys. Rev. Lett. 110, 156402 (2013).
[Crossref] [PubMed]

2012 (1)

C. Ohm, C. Stampfer, J. Splettstoesser, and M. R. Wegewijs, “Readout of carbon nanotube vibrations based on spin-phonon coupling,” Appl. Phys. Lett. 100, 143103 (2012).
[Crossref]

2011 (2)

J.R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: group theoretic approach,” New J. Phys. 13, 025025 (2011).
[Crossref]

M. W. Doherty, N. B. Manson, P. Delaney, and L. C. L. Hollenberg, “The negatively charged nitrogen-vacancy centre in diamond: the electronic solution,” New J. Phys. 13, 025019 (2011).
[Crossref]

2010 (5)

L. Robledo, H. Bernien, V. D. S. Toeno, and R. Hanson, “Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond,” New J. Phys. 13, 025013 (2010).
[Crossref]

M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Continuous quantum non-demolition measurement of Fock states of a nanoresonator using feedback-controlled circuit QED,” Phys. Rev. B 82, 094511 (2010).
[Crossref]

L. G. Zhou, L. F. Wei, M. Gao, and X. B. Wang, “Strong coupling between two distant electronic spins via a nanomechanical resonator,” Phys. Rev. A 81, 042323 (2010).
[Crossref]

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

2009 (2)

Z. Y. Xu, Y. M. Hu, W. L. Yang, M. Feng, and J. F. Du, “Deterministically entangling distant nitrogen-vacancy centers by a nanomechanical cantilever,” Phys. Rev. A 80, 022335 (2009).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

2008 (3)

P. Rabl, P. Cappellaro, M. V. G. Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, “Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,” Phys. Rev. B 79, 041302 (2008).
[Crossref]

O. Matsuda, O. B. Wright, D. H. Hurley, V. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[Crossref]

L. J. Rogers, S. Armstrong, M. J. Sellars, and N. B. Manson, “Infrared emission of the NV centre in diamond: Zeeman and uniaxial stress studies,” New J. Phys. 10, 103024 (2008).
[Crossref]

2007 (3)

D. F. V. James and J. Jerker, “Effective hamiltonian theory and its applications in quantum information,” Can. J. Phys. 85, 625–632 (2007).
[Crossref]

Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007).
[Crossref]

S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deléglise, U. B. Hoff, M. Brune, J. M. Raimond, and S. Haroche, “Quantum jumps of light recording the birth and death of a photon in a cavity,” Nature 446, 297–300 (2007).
[Crossref] [PubMed]

2006 (2)

X. Baia, T. A. Eckhausea, S. Chakrabartib, P. Bhattacharyab, R. Merlina, and C. Kurdak, “Phonon detection using quasi one-dimensional quantum wires,” Physica E 34, 592–595 (2006).
[Crossref]

N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
[Crossref]

2004 (1)

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent Shear Phonon Generation and Detection with Ultrashort Optical Pulses,” Phys. Rev. Lett. 93, 095501 (2004).
[Crossref] [PubMed]

2003 (1)

L. M. Duan, A. Kuzmich, and H. J. Kimble, “Cavity QED and quantum-information processing with “hot” trapped atoms,” Phys. Rev. A 67, 032305 (2003).
[Crossref]

2001 (1)

L. M. Duan, M. D. Lukin, I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[Crossref] [PubMed]

2000 (1)

J. H. Chai and Y. Q. Lu, “Effects of phase mismatch and losses on phonon squeezing and quantum nondemolition measurements in detection of hypersonic phonons by squeezed light,” Physica B 291, 292–298 (2000).
[Crossref]

1999 (1)

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[Crossref]

1998 (1)

P. Grangier, J. A. Levenson, and J. P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
[Crossref]

1997 (1)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[Crossref]

Amezcua, M.

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016).
[Crossref] [PubMed]

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D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016).
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D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016).
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Wang, R. X.

K. Cai, R. X. Wang, Z. Q. Yin, and G. L. Long, “Second-order magnetic field gradient-induced strong coupling between nitrogen-vacancy centers and a mechanical oscillator,” Sci. China Phys. Mech. 60, 070311 (2017).
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Wang, X. B.

L. G. Zhou, L. F. Wei, M. Gao, and X. B. Wang, “Strong coupling between two distant electronic spins via a nanomechanical resonator,” Phys. Rev. A 81, 042323 (2010).
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C. Ohm, C. Stampfer, J. Splettstoesser, and M. R. Wegewijs, “Readout of carbon nanotube vibrations based on spin-phonon coupling,” Appl. Phys. Lett. 100, 143103 (2012).
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M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Continuous quantum non-demolition measurement of Fock states of a nanoresonator using feedback-controlled circuit QED,” Phys. Rev. B 82, 094511 (2010).
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O. Matsuda, O. B. Wright, D. H. Hurley, V. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
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O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent Shear Phonon Generation and Detection with Ultrashort Optical Pulses,” Phys. Rev. Lett. 93, 095501 (2004).
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Z. Y. Xu, Y. M. Hu, W. L. Yang, M. Feng, and J. F. Du, “Deterministically entangling distant nitrogen-vacancy centers by a nanomechanical cantilever,” Phys. Rev. A 80, 022335 (2009).
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Y. Yanay and A. A. Clerk, “Shelving-style phonon-number detection in quantum optomechanics,” New J. Phys. 19, 033014 (2017).
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Yang, W. L.

Y. Ma, Z. Q. Yin, P. Huang, W. L. Yang, and J. F. Du, “Cooling a Mechanical Resonator to Quantum Regime by heating it,” Phys. Rev. A 94, 053836 (2016).
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Z. Yin, W. L. Yang, L. Sun, and L. M. Duan, “Quantum network of superconducting qubits through opto-mechanical interface,” Phys. Rev. A 91, 012333 (2015).
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K. Cai, R. X. Wang, Z. Q. Yin, and G. L. Long, “Second-order magnetic field gradient-induced strong coupling between nitrogen-vacancy centers and a mechanical oscillator,” Sci. China Phys. Mech. 60, 070311 (2017).
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Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007).
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Z. Yin, N. Zhao, and T. Li, “Hybrid opto-mechanical systems with nitrogen-vacancy centers,” Sci. China Phys. Mech. 58, 050303 (2015).
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L. G. Zhou, L. F. Wei, M. Gao, and X. B. Wang, “Strong coupling between two distant electronic spins via a nanomechanical resonator,” Phys. Rev. A 81, 042323 (2010).
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S. Saito, X. Zhu, R. Amsüss, Y. Matsuzaki, K. Kakuyanagi, T. Shimo-Oka, N. Mizuochi, K. Nemoto, W. J. Munro, and K. Semba, “Towards realizing a quantum memory for a superconducting qubit: storage and retrieval of quantum states,” Phys. Rev. Lett. 111, 107008 (2013).
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D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016).
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Figures (4)

Fig. 1
Fig. 1 (a) A schematic diagram of the phononic crystal. The NVE located near the surface coupling to a single phonon and a laser field. (b) The electronic structure of the NV center. Ω is the Rabi frequency between the energy levers |0〉 and | + 1〉 induced by the microwave drive. ωm is the phonon mode. ωo is the optical driving frequency between the energy levels |0〉 and |E〉 inducing the Rabi frequency of Ωo.
Fig. 2
Fig. 2 The situation of the symmetric output phonon shape. The red line is the driving pulse shape applying to the NVE, and the black line represents the pulse shape of the output phonon.
Fig. 3
Fig. 3 The schematic diagram of the phonon resonance spectrum with waist width of γe. The vertical coordinate represents the probability of a single phonon being absorbed. Nph represents the number of the absorbed phonon. The red line shows that, when there is one single phonon absorbed, the resonance frequency will shift distinctively compared with the non-phonon-absorbed situation.
Fig. 4
Fig. 4 The numerical simulation results. (a) is the absorbing process. The black lines represent the process with γ0/2π = 0.16kHz, γ1/2π = 0.16kHz, γm/2π = 0.16kHz, gm/2π = 0.96MHz, and κ/2π = 0.32MHz. The blue lines show the process with γ0/2π = 1.6kHz, γ1/2π = 1.6kHz, γm/2π = 1.6kHz, gm/2π = 0.96MHz, and κ/2π = 0.32MHz. The red ones denote the process with γ0/2π = 0.16kHz, γ1/2π = 0.16kHz, γm/2π = 0.16kHz, gm/2π = 1.92MHz, and κ/2π = 0.64MHz. (b) is the emitting process, where the black line represents the input phonon shape, and the red, blue and green lines represent the output phonon shapes when the coupling strength are at gm/2π = 0.96MHz, 0.64MHz and 0.32MHz respectively with the same value of κ = 0.32MHz. The overlap between the input pulse and the output pulse are 98.57%, 98.32% and 94.38% for the red, blue and green lines respectively.

Equations (32)

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H = i = 1 N G m ( | + 1 i 1 | a m + | 1 i + 1 | a m ) + Ω ( t ) 2 i = 1 N ( | 0 i + 1 | + | + 1 i 0 | ) + i κ / 2 π + Δ ω e + Δ ω e d ω [ a m e ( ω ) a m e ( ω ) ] + + Δ ω e + Δ ω e d ω [ ω e ( ω ) e ( ω ) ] .
H = i = 1 N G m ( | + 1 i 1 | a m + | 1 i + 1 | a m ) + Ω ( t ) 2 i = 1 N ( | 0 + 1 | + | + 1 0 | ) .
H = g m ( a a m + a a m ) + Ω ( t ) 2 ( d + d ) .
H = [ 0 g m 0 g m 0 Ω ( t ) 2 0 Ω ( t ) 2 0 ] .
H = g m ( a a m + a a m ) + Ω ( t ) 2 ( d + d ) + i κ / 2 π Δ ω e + Δ ω e d ω [ a m e ( ω ) a m e ( ω ) ] + Δ ω e + Δ ω e d ω [ ω e ( ω ) e ( ω ) ] .
| Ψ = c d | D | ϕ 0 + | N 1 | 0 + 1 | 0 0 | 0 m | ϕ 0 ,
| ϕ 1 = Δ ω e + Δ ω e d ω c ω e ( ω ) | ϕ 0 ,
c ˙ d = ( κ / 2 π cos θ ) Δ ω e + Δ ω e c ω d ω ,
c ˙ ω = i ω c ω + κ / 2 π c d cos θ .
c ω ( t ) = κ / 2 π 0 t e i ω ( t τ ) c d ( τ ) cos θ ( τ ) d τ ,
c ˙ d = κ cos θ π 0 t sin [ δ ω ( t τ ) ] ( t τ ) c d ( τ ) cos θ ( τ ) d τ .
δ ( x ) = lim k 1 π sin k x x ,
c ˙ d = κ 2 c d ( t ) cos 2 θ .
c d = e κ 2 0 t cos 2 θ ( τ ) d τ ,
c ω ( t ) = κ / 2 π 0 t e i ω ( t τ ) e κ 2 0 t cos 2 θ ( τ ) d τ c o s θ ( τ ) d τ .
f ( t ) = 1 2 π Δ ω e + Δ ω e d ω c ω ( T ) e i ω ( t T )
= κ cos θ ( t ) e κ 2 0 t cos 2 θ ( τ ) d τ .
f ( t ) = κ 1 + exp ( κ T 2 ) exp [ κ ( t T 2 ) / 2 ] + exp [ κ ( t T 2 ) / 2 ] ,
H = ω m a m a m + ω 0 E | E E | + Ω o 2 ( e i ω o t | 0 E | + e i ω o t | E 0 | ) + g ( a m + a m ) | E E | .
U = exp [ g ω m ( a m a m ) | E E | ]
H ˜ = U H U = ω m a m a m g 2 ω m | E E | + ω 0 E | E E | + Ω o 2 [ e i ω o t g ω m ( a m a m ) | 0 E | + H . c . ] .
H ˜ r = δ | E E | + g Ω o 2 ω m a m | E 0 | + g Ω o 2 ω m a m | 0 E | ,
H ˜ r = g Ω o 2 ω m e i δ t a m | E 0 | + g Ω o 2 ω m e i δ t a m | 0 E | .
H ˜ e f f = g 2 Ω o 2 4 ω m 2 δ a m a m ( | E E | | 0 0 | ) g 2 Ω o 2 4 ω m 2 δ | 0 0 | .
Δ f s = g 2 Ω 0 2 2 ω m 2 δ .
H c = i γ 1 2 | N 1 1 | 1 + 1 | 0 0 | 0 m i γ 0 2 | N 1 1 | 0 + 1 | 1 0 | 0 m i γ m 2 a m a m + g m ( a a m + a a m ) + Ω ( t ) 2 ( d + d ) + i κ / 2 π Δ ω e + Δ ω e d ω [ a m e ( ω ) a m e ( ω ) ] + Δ ω e + Δ ω e d ω [ ω e ( ω ) e ( ω ) ] .
H c = i γ 1 2 | N 1 1 | 1 + 1 | 0 0 | 0 m i γ 0 2 | N 1 1 | 0 + 1 | 1 0 | 0 m i γ m 2 a m a m + g m ( a a m + a a m ) + Ω ( t ) 2 ( d + d ) + i κ e j = 1 n δ ω [ a m e j a m e j ] + j = 1 n ( ω j δ ω e j e j ) ,
| Ψ > = ( c 1 | N 1 | 0 + 1 | 0 0 | 1 m + c 2 | N 1 1 | 1 + 1 | 0 0 | 0 m + c 3 | N 1 1 | 0 + 1 | 1 0 | 0 m ) | ϕ 0 + | N 1 | 0 + 1 | 0 0 | 0 m | ϕ 1 .
c ˙ 1 = γ m 2 c 1 i g m c 2 + κ e j = 1 n b ˜ j ,
c ˙ 2 = γ 1 2 c 2 i g m c 1 i Ω ( t ) 2 c 3 ,
c ˙ 3 = γ 0 2 c 3 i Ω ( t ) 2 c 2 ,
b ˙ j = κ e c 1 i b ˜ j ω j ,

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